Method for determination of roll density

ABSTRACT

A method for determining roll density in connection with a web reel. A known relation between the length (l), diameter (D), basis weight (b) and density (ρ) of the wound roll, formula (1) is used. Distortions caused in the measurement results by noise and other disturbances are eliminated by using a recursive time variant least squares method in the processing of the measurement results l=πρ(D 2 −D 2   0 )/4b.

CROSS REFERENCES TO RELATED APPLICATIONS

This application is a U.S. National Stage application of InternationalApplication No. PCT/FI02/00464, filed May 30, 2002, and claims priorityon Finnish Application No. 20011274, Filed Jun. 15, 2001, the disclosureof which is incorporated by reference herein.

STATEMENT AS TO RIGHTS TO INVENTIONS MADE UNDER FEDERALLY SPONSOREDRESEARCH AND DEVELOPMENT

Not applicable.

BACKGROUND OF THE INVENTION

The invention concerns a method for determination of roll density, inwhich method the density of the web to be wound is determined inconnection with the web reel and which method uses the known relationbetween the length, diameter, basis weight and density of the web to bewound ${l = \frac{{\pi\rho}\left( {D^{2} - D_{0}^{2}} \right)}{4b}},$

-   -   l=length of web to be wound    -   π=3.1415926 . . .    -   ρ=density    -   D=diameter    -   D_(o)=diameter of winding core    -   b=basis weight

A calculation formula is known in the state of the art, which can beused e.g. to determine the ply thickness of the wound web or to measurethe roll density.

The publication Roisum, D. R. “The Measurement of Web Stresses duringRoll Winding”, Oklahoma State University, 1990, s. 140-141 presents aformula in connection with measurement of roll density:${\rho = \frac{4{b({length})}}{\pi\left( {d_{i}^{2} - d_{i - 1}^{2}} \right)}},$

-   -   ρ=average web density over the measured distance    -   π=3.1415926 . . .    -   length=cumulated web length over the measured distance    -   d_(i)=roll diameter on measurement occasion i    -   b=basis weight of web

The formula was used in FI patent application 780893 for calculation ofthe thickness of a paper web:$d = \frac{R_{k} - R_{k - 1}}{n_{k} - n_{k - 1}}$  d(L _(k) −L_(k−1))=π(R _(k) ² −R _(k−1) ²),

-   -   d=average web thickness over the measuring distance    -   R_(k)=roll radius on measurement occasion k    -   L_(k)=web length on measurement occasion k    -   π=3.1415926 . . .    -   n_(k)=cumulative number of plies

A similar formula is also presented in the publication Happonen, E.“Paperirullan Rullaustiheyden Mittauslaitteisto” (“Equipment forMeasuring the Winding Density of a Paper Roll” Diploma Work),Diplomityö, Teknillinen Korkeakoulh (University of Technology), 1985 p.17, which presents a calculation formula for the average thickness ofwound paper in connection with the measurement of the thickness of woundpaper over a certain winding distanceD_(k)=(R_(k)−R_(k−1))/(n_(k)−n_(k−1)).

However, it has proved problematic in the determination of web densityor thickness to eliminate the inexactness resulting in measurementresults from noise and from other disturbances.

As is known in the state of the art, the formula has not been usedgenerally, because the density measurement according to the formula hasbeen prevented by the inexact diameter measurement and by the attendingvibration frequency of the roll center. This is especially problematicwhen measurement of the location of the roll center is used in thediameter measurement.

As regards the state of the art, reference is also made to DE patentpublication 41 28 706, which presents a method for determination of rolltightness when winding a material web on a winding machine, where thethickness of wound plies is found out for the roll tightness and whereinthe length of the wound web affecting the diameter is measured directly.In the method, the roll diameter is measured directly by degrees as atransition of the roll center and then the ply thickness is calculatedusing a formula similar to the one described above${S = \frac{\left( {D_{2}^{2} - D_{1}^{2}} \right)\pi}{4\Delta\quad l}},$

-   -   S=average web thickness over the measuring distance    -   D₁=roll diameter in the beginning of the measuring distance    -   D₂=roll diameter at the end of the measuring distance    -   π=3.1415926 . . .    -   Δl=web length cumulated over the measuring distance

In the calculation, high-frequency disturbances caused in the diametermeasurement by vibration of the roll center are eliminated by low-passfiltration. In this known solution the filtration is thus based on theassumption that disturbances are of a high frequency and the filtrationwill thus be fairly rough.

The use of low-pass filtration to reduce the share of the noise of themeasurement signal is based on the assumption that the noise summed onthe measurement is of zero average value, that is, unbiased, and thatits frequency content differs from the measurement signal proper. Thefilter removes from the measurement those higher frequencies, which thenoise brings along, whereby the desired original measurement signal willremain. This may cause inexactness, since also a part of the desiredmeasurement signal is filtered and also a part of the noise will remain.In addition, phase lag, that is, delay, may result in the measurementsignal.

If low-pass filtration were to be used in order to achieve an efficientfiltration at all machine ruling speeds and with all roll diameters,then the boundary frequency of low-pass filtration ought to be changedconstantly as these factors are changing. Since the main reason formeasurement noise is the waving in diameter measurement caused by theeccentricity of the roll center, the frequency content of measurementnoise is strongly dependent on the rotation frequency of the roll, whichagain depends on the running speed and on the roll diameter, beingtypically at a maximum a little while after winding has begun. It is aproblem with the state-of-the-art procedure that it is not easy inpractice to implement a constant changing during operation of theboundary frequency of any higher rate low-pass filter.

Various ways of measuring the roll diameter are known in the state ofthe art, the most widely used being pulse measurement, wherein pulsemeasurement is used for measuring the roll circumference. The diameterinformation obtained through pulse measurement has been used in order tofind out the web thickness. When one ply is completed on the roll, thelength measure is taken-from the carrying roller and the paper thicknessis obtained by finding out how quickly the roll diameter increases. Itis problematic to determine the web thickness exactly, because whenwinding large-diameter rolls wherein the web is thin, differences inorders of magnitude have caused inexactness.

The roll hardness determines how tightly the roll is wound. Thiscorresponds with a certain internal compression pressure distributionbetween the plies, which is the higher the tighter the roll. Inaddition, roll hardness depends on the hardness of the paper itself,that is, on the elastic modulus in the Z direction, which is differentfor different paper grades, that is, rolls of various kinds wound to thesame tightness may have a different hardness. Roll density correlateswith tightness or hardness, because an increased compression pressurewill cause a deformation that will compress the paper layers together.

SUMMARY OF THE INVENTION

The invention aims at bringing about a method more exact than thedensity measuring methods known in the state of the art. A particularobjective of the invention is to bring about a method, whereindistortions caused by noise and other disturbances in the measurementresults are eliminated.

In order to achieve the objectives presented above and those emerginghereinafter the method according to the invention is mainlycharacterized in that in the method distortions caused by noise and byother disturbances in the measurement results are eliminated by using inthe processing of measurement results a recursive time variant leastsquares method.

In the method according to the invention, a value is preferablydetermined for “filtration” based on measurement and mathematicalstatistics and calculus of probability are preferably applied, wherebyan exact measurement results is attained, when that measurement noise iseliminated, which is mainly caused by oscillation of the roll center.

In an advantageous embodiment of the method for density measurementaccording to the invention, variables are initialized by two pointsdetermined by the nominal density before the first measurement. Thecorrelation matrix and regression vector are then updated for eachfollowing measurement. The correlation matrix is then reversed,whereupon the paper thickness may be calculated. Based on the paperthickness the density is calculated and the error variance is updated,based on which the confidence limit is calculated, for example, for a95% probability, and if the confidence limit is too big, the bufferlength is increased, while if the confidence limit is too short thebuffer length is decreased. The speed of oblivion is then determined andthe following measurement is carried out after a chosen web length, forexample, when the web length has increased by 1 meter.

The method according to the invention uses a recursive time variantleast squares method, which is easy to implement in program terms andgives the density value directly and does not cause any distortions inthe shape of the density curve. To the least squares method a method ofmathematical statistics can be applied, with the aid of which a relationis obtained between exactness of measurement and the method's built-infiltration constant, whereby statistical confidence limits aredetermined for the measurement value, that is, the probability, by whichthe measurement value is closer to the correct value by a certain limit.In this manner any inaccuracies caused by noise and other disturbancesin the measurement results are controlled in such a way that nodistortions will occur in the measurement results.

Thus, the method according to the invention is based on a statisticalmethod, which is not dependent on any frequency differentiation ofsignals, whereby filtration of measurement results is not performed in aseparate stage, but if required the effect of filtration is estimatedthrough the confidence limit, the value of which is directly related tothe measurement signal proper. In accordance with an advantageousfeature of the method according to the invention, the program itselfdetermines a suitable filtration constant continuously based on thenoise of the measurement.

In connection with the method according to the invention the diametercan be measured as a pulse measurement diameter in connection withwinding-in, as a diameter measured from the location of the roll center,by a distance meter, for example, a laser meter or any other suitablemeasuring procedure that is sufficiently accurate. In unwinding, thelocation of the roll center does not change, so the diameter measurementis most suitably performed as a pulse measurement or using an ultrasonicdistance meter from atop the roll.

In connection with the invention it is advantageous to use a linearsensor in measuring the tightness of winding-in separately for eachstation and based on the diameter and web length given by the linearsensor.

In the method according to the invention, the diameter may be measuredin the desired manner, for example, by ultrasound, by a laser sensor,using which a measurement without contact is preferably achieved.

According to an advantageous feature of the method according to theinvention, the determination of the measurement buffer length is alsocarried out automatically, whereby the measurement adapts to the varyingnoise and preserves its accuracy.

The method according to the invention thus utilizes a physical model inprocessing the measurement. When using a least squares procedure in thismethod according to the invention, this advantage is obtained, that is,an unbiased estimate of the density is obtained from the measurementdata.

According to an advantageous additional feature of the method accordingto the invention, the measurement buffers are initialized in such a waythat the measurement first shows the density value given by the user orthe density value measured in the beginning of the previous winding,whence it begins following the measured value, as the roll startsrotating. This speeds up penetration of the initial transient and themeasurement is made to begin as early as possible.

An even more exact initialization is achieved by first storing asuitable quantity of measurement data and by calculating the densitybackwards towards the smaller diameter from this data, whereby a veryexact initial value is obtained for the roll bottom. However, this istougher in terms of calculation.

In accordance with an advantageous application of the invention, themeasuring method according to the invention may be extended further byapplying an Extended time variant Kalman Filter. The purpose of theKalman Filter is to utilize not only the direct measurement but also aknown or estimated dynamic model of the system, which includes aphysical description of the system known beforehand:{dot over (x)}=f(x(t),u(t),t)+w(t)z(t)=h(x(t),u(t),t)+v(t)

In this description x is the state vector, which defines the interiorstate of the system. In a winding model components of the state vectorare e.g. the radial compression pressure and tangential tensile stressof the roll's surface ply. Vector u includes control magnitudes, such ase.g. the nip load, web tension, winding power, running speed. Thecontrol magnitudes are known or they can be measured directly. Vectors wand v are noises disturbing the system and the measurement. Vector z isa measurement magnitude, in this case the density. Function f is adescription of the winding model, it tells how the roll's internalstress distribution results from the effect of the control magnitudes(for example, Jorkama, M. “Contact Mechanical Model for Winding Nip”,Teknillinen Korkeakouhlu, 2001 and Hakiel, Z. “Nonlinear Model for WoundRoll Strees”, Tappi Journal, 1987). The explicit time variable tells theeffect on it by magnitudes not mentioned separately, such as e.g. theroll diameter and the paper thickness of the arriving web. Function h isa description of pressures through roll deformation on the density. Itis dependent on the paper characteristics, such as the radial andtangential elastic modulus and the friction. The model may also bestatic, if dynamics are of no significance (slow changes of ratings):x(t)=f(u(t),t)+w(t)

The idea of the Kalman filter is to form a reverse description h⁻¹,which is used to measure the roll pressure distribution indirectly, whenthe measured density and the dynamic physical model describing thewinding are known. Forming the extended time variant Kalman filter isknown in the art beginning from the system description. The modelstructure presented herein is by no means the only one that can beapplied to winding, nor is the intention to be limited to it only.

The method according to the invention gives by little loading of theprocessor and light calculation and with easy implementation in aprogram such a result in an optimum manner in terms of mathematicalstatistics, which is very easy to set up for use with different papergrades.

BRIEF DESCRIPTION OF THE DRAWINGS

In the following, the invention will be described in greater detail withreference to the figures shown in the appended drawing, wherein

FIGS. 1A and 1B are schematic block views of the density measurementaccording to the invention, and

FIG. 2 is a schematic view of an example comparing the previously knownpulse density measurement and the measuring method according to theinvention.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

The variables shown in the formulas in FIGS. 1A and 1B have thefollowing meanings:

D₀ = initial diameter D = measured roll diameter l = measured web lengthin roll b = basis weight given from screen ρ = nominal density givenfrom screen λ = oblivion parameter (<1) φ = regression vector =[D²1]^(T) Y = measurement = 1 θ = parameter vector, whose firstcomponent is inversely proportional to the paper thickness

In the advantageous application of the density measurement methodaccording to the invention shown in FIGS. 1A and 1B before the firstmeasurement occasion, measurement occasions, block 11, the variables areinitialized by two points determined by the nominal density, block 12,formula: $l = \frac{{\pi\rho}\left( {D^{2} - D_{0}^{2}} \right)}{4b}$

Thereafter, on each measurement occasion the correlation matrix and theregression vector are updated, blocks 13, 14, formulas:$R_{n} = {{\Phi_{n}^{T}\Phi_{n}} = {{{{\lambda\Phi}_{n + 1}^{T}\Phi_{n + 1}} + {\phi_{n}\phi_{n}^{T}}} = {{\lambda\quad R_{n - 1}} + \begin{bmatrix}D^{4} & D^{2} \\D^{2} & 1\end{bmatrix}}}}$$\phi_{Yn} = {{{\lambda\phi}_{{Yn} - 1} + {\phi_{n}Y_{n}}} = {{\lambda\phi}_{{Yn} - 1} + \begin{bmatrix}{D^{2}l} \\l\end{bmatrix}}}$

Thereafter, the correlation matrix is reversed, whereupon the paperthickness can be calculated, blocks 15, 16, formulas:$P_{n} = {R_{n}^{- 1} = {\frac{1}{detR}\begin{bmatrix}R_{22} & {- R_{12}} \\{- R_{12}} & R_{11}\end{bmatrix}}}$  (θ_(n)=P_(n)θ_(Yn))${paper\_ thickness} = \frac{\pi}{4\left( {{P_{11}\phi_{Y1}} + {P_{12}\phi_{Y2}}} \right)}$

Based on the paper thickness, the density is calculated, block 17,formula:Density=Basis_Weight/Paper_Thicknessand the error variance is updated, block 18, formula:ε_(n) =Y _(n)−θ_(n)φ_(n)Error Variance_(n)=α·ErrorVariance_(n-1)+(1−α)ε_(n) ²in which context the confidence limit is calculated, for example, for aprobability of 95%, block 19, formula: $\begin{matrix}{{ConfidenceLimit}_{n} = {1.739606726\quad\bullet}} \\{{Basis\_ Weight}\frac{4}{\pi}\sqrt{{ErrorVariance}_{n}P_{11}}}\end{matrix}$and if the confidence limit is too large, the buffer length isincreased, whereas if the confidence limit is too short, the bufferlength is decreased, blocks 20, 21. Thereafter the speed of oblivion isdetermined, block 22, formula:$\lambda = \frac{1 - {BufferLength}}{BufferLength}$and the following measurement is performed after a chosen web length,for example, when the web length has increased by 1 meter, block 23.

FIG. 2 is a schematic view of an example comparing the previously knownpulse density measurement and the measuring method according to theinvention. In the figure, reference number 31 indicates the curveachieved with pulse density measurement and reference number 32indicates the result curve achieved with the density measurementaccording to the method in accordance with the invention. As can be seenin FIG. 2, the method according to the invention, gives an exacter andmore reliable result than the previously known density measurementmethod.

In the foregoing the invention was described by referring to its oneadvantageous application example only, but the intention is not to limitthe intention in any way strictly to the details of that example.

1. In a method for determination of roll density, in which density of awound roll formed of a web is determined at a multiplicity of occasionswhile reeling the web by measuring a parameter corresponding to at leastone value selected from the group consisting of: measured web length inthe roll (l), measured roll diameter (D), and basis weight of the web(b); and using a known relation between variables: measured web lengthin the roll (l), measured roll diameter (D), initial rolldiameter(D_(o)), basis weight of web (b) and density (ρ) of the woundroll; $l = \frac{{\pi\rho}\left( {D^{2} - D_{0}^{2}} \right)}{4b}$ theimprovement comprising: eliminating distortions in said measuredparameter caused by noise and other disturbances, by using a recursivetime variant least squares method in the processing of said measuredparameter.
 2. The method of claim 1 wherein a method of mathematicalstatistics is applied to the least squares method, whereby a relation isobtained between the at least one measured value exactness and themethod's built-in elimination of noise and disturbances.
 3. The methodof claim 1 wherein statistical confidence limits are determined for theat least one measured value.
 4. The method of claim 1, wherein beforebeginning a measurement occasion, initializing the at least one measuredvalue by two points determined by a nominal density or by a precedingmeasurement; and further comprising carrying out the steps of: updatinga correlation matrix and a regression vector; reversing the correlationmatrix to calculate a paper thickness; calculating density of the woundroll based on the paper thickness, followed by; updating an errorvariance, followed by; calculating a confidence limit for a selectedprobability; if the confidence limit is below a selected lower limit orabove a selected upper limit then changing a buffer length; determininga speed of oblivion based on the buffer length; and beginning anothermeasurement occasion after a selected web length has been wound orunwound from the wound roll.
 5. The method of claim 1 wherein therecursive time variant least squares method uses an extended timevariant Kalman filter for further processing of the measurement resultsand for measuring the internal stresses of the roll.
 6. The method ofclaim 1 wherein the roll diameter is determined by contact-freemeasurement.
 7. The method of claim 6 wherein the diameter is measuredby pulse measurement, ultrasound or a laser sensor.
 8. The method ofclaim 1 wherein the density is measured separately for each of aplurality of stations based on the roll diameter and web length given bya linear sensor.
 9. The method of claim 1 wherein the method is appliedin connection with winding-in of the web.
 10. The method of claim 1wherein the method is applied in connection with unwinding of the web.11. The method of claim 1 wherein the method is applied in connectionwith winding of a paper or board web.